Choosing λ, market class by market class

λ is the vesting fraction — the share of each stake that vests to the opposing book on arrival. λ = 0 is the classic parimutuel. λ = 1 is the pure mechanism. Everything in between is a blend, and the blend is a real setting rather than a hedge: the venue picks it per market class and publishes it like a fee.

The reason this article exists is that the interior is not a compromise between two good things. It is a specific trade with a specific casualty, and the casualty is not obvious until you look at the table.

How the blend works

Vest a fraction λ of each stake by Rule 1; settle the remaining 1 − λ as a classic terminal pool. The composition with the capacity rule is part of the specification and is easy to get wrong:

Acceptance, capacity, and the seed clamp operate on the full stake exactly as at λ = 1. Only the settlement of the accepted amount splits.

Because vesting is linear and every stake vests the same fraction, a winner's blended payout collapses to a closed form from a single λ = 1 settlement:

Π_i(λ)  =  λ · (s_i + v_i)  +  (1 − λ) · s_i · M

with M the classic multiple of the accepted pool. That is how the reference implements it, and how the table below is generated — one settlement, then a blend, not two settlements.

The trilemma

Four properties. No point on the dial delivers all four.

The word doing the work is void in the middle row's second column. Not "reduced." Not "weaker." The creation floor does not exist at any interior λ, because the classic component pays the seeder less than its stake whenever its realized-side share falls below its pool share.

Measured, across 40,000 markets:

Even λ = 0.75 — which sounds like "almost all the way there" — dips to −3.0%. The floor is a knife edge at λ = 1, not a gradient.

The rest of the dial

Two structural facts about how this degrades.

The late entrant's reward is exactly linear in λ. That is P7: a late entrant's multiple is exactly λ·1 + (1−λ)·M. The sniper's own multiple runs 1.273× to 1.005× linearly across those same rows, by the composition identity.

Dilution degrades better than linearly. The measured mean sits strictly below the linear reading at every interior λ. So the dial protects early capital better than a naive reading suggests — a pleasant surprise, with a caveat.

The caveat is that the per-position dilution law is signed, not one-sided. Position i's dilution satisfies D_i(λ) ⋚ (1−λ)·D_i(classic) according as the position's pure multiple 1 + v_i/s_i is above or below the classic multiple M. An individual early position on the wrong side of that crossover can exceed the naive per-position bound, and the suite pins a log where exactly that happens. So "dilution is about (1−λ) of classic" is true in aggregate and false for some positions, and a venue should not promise it per position.

The market-class table

Reading the interior honestly

The interior of the dial is not "most of the benefits, some of the safety." It is:

• You still need a lock window, because the buzzer snipe pays λ + (1−λ)·M and that is positive at every λ < 1. The reason to adopt has partly evaporated.
• You have no creation floor, so the whole seeding-as-a-float argument does not apply, and your seeders are back to bearing a real expected loss.
• You get partial hedging and partial dilution protection, which is genuinely useful if hedging is what you needed.

So the interior is the right answer for exactly one situation: markets where hedging demand is the point. §14 states this as its own conclusion — "a hedging market is an operator-seeded or floor-waived market" — and calls it "the honest reason λ exists."

There is a second-order reason to care about hedgers beyond serving them: hedging demand is a principal source of the uninformed flow that the secondary layer needs in order to exist. Push all of it out with λ = 1 and you have made the late-price problem harder for yourself.

Using λ as a migration path rather than a destination

The most useful thing about the dial is that it lets you move without a cutover. You are at λ = 0 today. You can run a class at 0.25, watch what your users do, and move again — and the paper publishes what every stop costs, so nothing about the trip is a surprise.

But do not park in the interior thinking it is a safe middle. The two ends are the coherent positions:

• λ = 0: full late trading, full hedging, no early reward, needs a lock. Your current venue.
• λ = 1: no late free-ride, exact dilution protection, unconditional floor, no exit and no hedge, needs a secondary layer.

The interior has the lock and no floor. It is a waypoint, or it is a deliberate choice made for hedgers. It is not a default.

What λ does not touch

Worth stating because it removes a class of worry:

• Acceptance and capacity operate on the full stake at every λ. A partial fill at λ = 0.5 is the same partial fill it would be at λ = 1.
• Conservation holds at every λ, for every log and outcome, exactly, in integer units.
• The seed clamp — the joint vintage-0 acceptance condition a_o ≤ κ · min_{w≠o} a_w — is λ-independent.
• Settlement cost stays O(1); the blend is one closed-form combination applied to a single λ = 1 settlement.

And one implementation note: λ = 0 under this mechanism is not quite the classic parimutuel. It differs by Rule 2's refusal, measured at 0.10% of gross in the study. If you are running an A/B against your existing pools, that is the discrepancy you will see, and it is expected.

Next: Implementing it — the settler, the arithmetic, and the suite.